基于Hausdroff分形维数的碳酸盐岩酸蚀蚓孔计算模型

在碳酸盐岩基质酸化过程中,一个主要特征是形成蚓孔。欧几里得空间认为蚓孔是无序的、杂乱无章的,采用经典数学的方法进行模拟难度极大。为了能够对酸蚓孔进行精确的定量分析、优化酸液用量、施工排量,以期获得最优酸化效果,需要建立蚓孔计算模型。由于蚓孔的整体与局部具有相似性以及标度不变性,因此利用分数维描述蚓孔,引入“数盒子”计算法对蚓孔的Hausdroff分形维数进行研究,推导了碳酸盐岩基质酸化蚓孔有效作用距离的分形模型。进行了影响参数敏感性分析及现场酸化实例计算,结果表明酸化后表皮系数与计算结果较为一致。对酸化优化设计具有较好的指导意义。

Wormhole Calculation Model of Carbonate Acidizing Based on Hausdroff Fractal Dimension

The main characteristic of carbonate acidizing is the formation of wormhole. The shape of wormhole was thought as out of order in Euclidean space. As the randomness of acid wormhole, it is difficult to simulate acidizing with classical mathematics. In order to analyze wormhole quantitatively, optimize assumed acid volume, pumping rate and get best production rate, we need to establish calculation model of wormhole. Because wormhole meets the qualification of fractal, the method of fractal can be used to describe it. Hausdroff fractal dimension was calculated with box counting method. Multi-fractal analysis for radius acid wormhole and vertical acid pattern were studied with typical range of 1.6-1.7 based on different acidizing parameters. Main parameters influenced on fractal dimensions of carbonate acidizing wormhole include H+ diffusion coefficient, injection rate, heterogeneity of formation porosity and fracture. Then a fractal model of carbonate acidizing wormhole effective length was deduced. Besides, the models of consumed acid volume, pumping rate and skin factor were given combined with acidizing experiments. Results showed that wormhole has fractal features under suitable pumping rate, and some key parameters can be optimized with this model established. Wormhole penetration length was calculated with